Av(123, 231)

Permutation examples

length 2: 12, 21

length 3: 132, 213, 312, 321

length 4: 1432, 2143, 3214, 4132, 4213, 4312, 4321

length 5: 15432, 21543, 32154, 43215, 51432, 52143, 53214, 54132, 54213, 54312, 54321

ATRAP tree (size 5, depth 3)

Legend

$\mathcal{A}$ = Av(123, 231)

$\mathcal{B}$ = Av(12)

Generating function

$A(x) = \frac{- 2 x^{2} + 2 x - 1}{x^{3} - 3 x^{2} + 3 x - 1}$

Coefficients

1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ...

System of equations

$\operatorname{F_{9}}{\left (x \right )} = \operatorname{F_{5}}{\left (x \right )} + \operatorname{F_{8}}{\left (x \right )}$

$\operatorname{F_{5}}{\left (x \right )} = 1$

$\operatorname{F_{8}}{\left (x \right )} = \operatorname{F_{6}}{\left (x \right )} + \operatorname{F_{7}}{\left (x \right )}$

$\operatorname{F_{6}}{\left (x \right )} = - \frac{x}{x - 1}$

$\operatorname{F_{7}}{\left (x \right )} = - \frac{x^{2}}{\left(x - 1\right)^{3}}$